Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups

Sunthrayuth Pongsakorn, Kumam, Kumam Poom

Description

In this paper, we introduce new iterative algorithms for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings and the set of common fixed points of a one-parameter nonexpansive semigroup in Banach spaces. Furthermore, we prove the strong convergence theorems of the sequence generated by these iterative algorithms under some suitable conditions. The results obtained in this paper extend the recent ones announced by many others. Mathematics Subject Classification (2010): 47H09, 47J05, 47J25, 49J40, 65J15

Sujets

• Iterative method
• Variational inequality
• Strong convergence
• Banach space

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 5
• Langue: English

Extrait

Sunthrayuth and KumamJournal of Inequalities and Applications2012,2012:133 http://www.journalofinequalitiesandapplications.com/content/2012/1/133

R E S E A R C HOpen Access Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups * Pongsakorn Sunthrayuth and Poom Kumam

* Correspondence: poom. [e-mail] Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Abstract In this paper, we introduce new iterative algorithms for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings and the set of common fixed points of a oneparameter nonexpansive semigroup in Banach spaces. Furthermore, we prove the strong convergence theorems of the sequence generated by these iterative algorithms under some suitable conditions. The results obtained in this paper extend the recent ones announced by many others. Mathematics Subject Classification (2010):47H09, 47J05, 47J25, 49J40, 65J15 Keywords:oneparameter nonexpansive semigroup, perturbed mapping, iterative algorithm, variational inequality, strong convergence, Banach space, common fixed points

1 Introduction Variational inequality theory has been studied widely in several branches of pure and applied sciences. Indeed, applications of variational inequalities span as diverse disci plines as differential equations, timeoptimal control, optimization, mathematical pro gramming, mechanics, finance, and so on (see, e.g., [1,2] for more details). Note that most of the variational problems include minimization or maximization of functions, variational inequality problems, quasivariational inequality problems, decision and management sciences, and engineering sciences problems. For more details, we recom mend the reader [38,2931]. LetXbe a real Banach space, andX* be its dual space. Theduality mapping ∗ X is defined by J:X→2    ∗2 J ( x ) =f∈X:x,f=||x||,||f||=||x||,

where〈∙, ∙〉denotes the duality pairing betweenXandX*. IfX: =His a real Hilbert space, thenJ=IwhereIis the identity mapping. It is well known that ifXis smooth, thenJis singlevalued, which is denoted byj(see [9]).

© 2012 Sunthrayuth and Kumam; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Sunthrayuth and KumamJournal of Inequalities and Applications2012,2012:133 http://www.journalofinequalitiesandapplications.com/content/2012/1/133

LetCbe a nonempty closed and convex subset ofXandTbe a selfmapping ofC. We denote®and⇀by strong and weak convergence, respectively. Recall that a map pingT:C®Cis said to beLLipschitzianif there exists a constantL >0 such that ||Tx−Ty|| ≤L||x−y||,∀x,y∈C. If 0 <L< 1, thenTis a contraction and ifL= 1, thenTis a nonexpansive mapping. We denote by Fix(T) the set of all fixed points set of the mappingT, i.e., Fix(T) = {xÎ C:Tx=x}. A mappingF:C®Xis said to beaccretiveif there existsj(xy)ÎJ(xy) such that   Fx−Fy,j x−y≥0,∀x,y∈C. A mappingF:C®Xis said to bestrongly accretiveif there exists a constanth> 0 andj(x  y)ÎJ(x  y) such that   2 Fx−Fy,j x−y≥η||x−y||,∀x,y∈C. Remark1.1. IfX: =His a real Hilbert space, accretive and strongly accretive map pings coincide with monotone and strongly monotone mappings, respectively. LetHbe a real Hilbert space, whose inner product and norm are denoted by〈∙, ∙〉and ||∙||, respectively. LetAbe a strongly positive bounded linear operator onH, that is, there exists a constant γ >0 such that 2  Ax,x  ≥ γ||x||,∀x∈H. (1:1) Remark1.2. From the definition of operatorA, we note that a strongly positive bounded linear operatorAis a ||A||Lipschitzian andhstrongly monotone operator. LetCbe a nonempty closed and convex subset of a real Banach spaceX. Recall that theclassical variational inequalityis to findx*ÎCsuch that    ∗ ∗ x,j x−x≥0,∀x∈C, (1:2) whereΨ:C®Xis a nonlinear mapping andj(xx*)ÎJ(xx*). The set of solution of variational inequality is denoted byVI(C,Ψ). IfX: =His a real Hilbert space, then (1.2) reduces to findx*ÎCsuch that   ∗ ∗ x,x−x≥0,∀x∈C. (1:3) A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH 1 min x∈C  Ax,x  −  x,u  , (1:4) 2 whereCis the fixed point set of a nonexpansive mappingTonHanduis a given point inH. In 2001, Yamada [10] introduced a hybrid steepest descent method for a nonexpan sive mappingTas follows: x n+1 =Tx n −µλ n F ( Tx n ) ,∀n≥0, (1:5)

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